The Square-Root GameAuthor(s): David BlackwellSource: Lecture Notes-Monograph Series, Vol. 35, Game Theory, Optimal Stopping, Probabilityand Statistics (2000), pp. 35-37Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/4356079 .

Accessed: 15/06/2014 04:09

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access toLecture Notes-Monograph Series.

http://www.jstor.org

This content downloaded from 91.229.229.203 on Sun, 15 Jun 2014 04:09:23 AMAll use subject to JSTOR Terms and Conditions

The Square-root Game

David Blackwell

University of California

Department of Statistics

367 Evans Hall # 3860

Berkeley, California 94720-3860

U. S. A.

e-mail:david.bl@stat.berkeley.edu

Abstract

In this work I give an elementary proof of the following : "The

absolute 1/2 moment of the beta (1/4,1/4) distribution about t is

independent of t for 0 < t < 1.

Keywords : beta distribution, two person game, Richardson extrapo-

lation.

Theorem 1 The absolute 1/2 moment of the beta (1/4,1/4) distribution

about t is independent of t for 0 < t < 1 :

/ p(x)(\x-t\{ll2))dx, Jo

where

p(x) = [x(l-z)](-3/4\

is independent oft for 0 < t < 1.

More generally, for any a with 0 < a < 1, the absolute a ? th moment of the beta ((1

? a)/2, (1

? a)/2) distribution about t is independent of t for

0<t<l.

This note, which I am pleased to write in honor of my old friend Tom

Ferguson, is about the process that led to the Theorem.

Quite a few years ago, shortly after Tom Ferguson got his first PC, I

asked him about the Square-root Game :

35

This content downloaded from 91.229.229.203 on Sun, 15 Jun 2014 04:09:23 AMAll use subject to JSTOR Terms and Conditions

36 David Blackwell

Square-root Game

Players I and II simultaneously choose numbers ? and y in the unit inverval.

Then II pays I the amount \x -

y\^l^2\

Later that day, Tom told me that the value of the game is .59907, to 5

places. I asked him how he got such accuracy, since he could solve games

only up to 30 ? 30 on his machine. He said that he'd used Richardson

extrapolation (which I'd never heard of). He told me a bit about Richardson

extrapolation, and we turned to other rhings.

Then, in my Fall 1994 game theory class I assigned, as a homework prob-

lem, to solve the Square-root Game to 3 places. Several students succeeded,

but one group of four students, working together, claimed to have solved

the game to 15 places. According to them, if either player used the beta

(1/4,1/4) strategy, then Player I's expected income, as calculated by Math-

ematica, was constant to 15 places, no matter what the other Player did.

They could not prove that a beta (1/4,1/4) strategy gave a constant income,

and neither could I.

Later I asked Jim Pitman about the more general case as stated in the

Theorem, and he gave a not-quite-elementary proof. Later he found a gen-

eralization to higher dimensions in Landkof [1972]. Finally I found an ele-

mentary proof, given below.

The method the four students used to get their solution is simple and

instructive.

1. They solved a discrete version, restricting each Player to the 21 choices

0, .05,..., .95,1. The good strategy for each player was a [/-shaped

distribution, symmetric about 1/2.

2. They calculated the variance of this distribution, and found the beta

distribution symmetric about 1/2 with the same variance. It was beta

(.2613, .2613).

3. They guessed that .2613 was trying to be .25, so tried beta(l/4,1/4) as a strategy.

Here is the proof of the Theorem. Fix a, 0 < a < 1, and put

/(*)= ?lv{x){\x-t\a)dx Jo

where p(x) = [x(l -

?)]^a+1^21

This content downloaded from 91.229.229.203 on Sun, 15 Jun 2014 04:09:23 AMAll use subject to JSTOR Terms and Conditions

The Square-root Game 37

We must show that / is constant on 0 < t < 1. Its derivative is

f'(t) = a [\(t -

??a-%(?) dx - a [ [(? -

?)(a_1)]?(?) dx

With the change of variable u = 1 ? ? in the second integral, we get

/'(t) = a[?(? -

?)(a~?)}?(?) dx- f ~\(1 - t -

u^-^IpCu) du]

= a(F(?)-F(l-?)), where

F(t) = /t[(i-x)(a-1)]p(x)dx. Jo

So we must show that F(l ?

t) = F(t). To evaluate F, make the lineax

fractional change of variable ? = (? ?

x)/(t ?

tx) : ? = t(l ?

^)/(l ?

tz) (see

Carr, [1970], Formula 2342). We get

F(t) = [i(l -

t)]?a-W [\(1 -

2)(-(a+1)/2)][^a"1)] dz jo

So F(l -1) = F(i), proving the Theorem.

So the value of the Square-root game is I's expected income when he chooses

? according to beta (1/4,1/4) and II chooses y = 0, namely

G(1/2)/G(1/4)G(1/4)) /1(x(1/2))([x(l-x)](-3/4))dx = G(1/2)G(3/4)G(1/4)

Jo = .599070117367796....

So Tom's first five places were correct.

Aknowledgement

I thank the referee for his kind remarks.

References

[1] CARR, G. S. (1970). Formulas and Theorems of Pure Mathematics.

Chelsea.

[2] LANDKOF, N. S. (1972). Foundations of Modern Potential Theory.

Springer-Verlag. N. S.

This content downloaded from 91.229.229.203 on Sun, 15 Jun 2014 04:09:23 AMAll use subject to JSTOR Terms and Conditions